Expand using Binomial Theorem {1+ x/2-2/x}^4, x not equal to 0 ( zero) ?

Question

Answer

Expanding {1 + x/2 - 2/x}⁴

The given expression is of the form (a+b)ⁿ, where a=1, b=x/2-2/x, and n=4. We can use the binomial theorem to expand it.

Binomial Theorem

The binomial theorem states that:

(a+b)ⁿ = ∑_k=0ⁿ C_kⁿ a^n-k b^k

where C_kⁿ is the binomial coefficient, given by:

C_kⁿ = n!/(k!(n-k)!)

Using this formula, we can expand the given expression:

Expansion

{1 + x/2 - 2/x}⁴ = ∑_k=0⁴ C_k⁴ 1^4-k (x/2-2/x)^k

Now we can simplify each term:

C₀⁴ 1^4-0 (x/2-2/x)⁰ = 1

C₁⁴ 1^4-1 (x/2-2/x)¹ = 4(x/2-2/x)

C₂⁴ 1^4-2 (x/2-2/x)² = 6(x/2)² + 6(2/x)² - 24

C₃⁴ 1^4-3 (x/2-2/x)³ = -4(x/2)³ + 32/x³

C₄⁴ 1^4-4 (x/2-2/x)⁴ = (x/2)⁴ + 16/x⁴ + 6(x/2)² - 48

Putting it all together:

{1 + x/2 - 2/x}⁴ = 1 + 4(x/2-2/x) + 6(x/2)² + 6(2/x)² - 24 - 4(x/2

Answer